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The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$?

The rather crude proof of Northcott's theorem bounds $\log{N(d)}$ by $O(d^2)$. On the other hand, upon extracting $\lfloor d/m \rfloor$-th roots from numbers of degree $m$ and logarithmic height $< d/m$, where $m$ is fixed but arbitrarily large, and using a generalization of Schanuel's theorem due to Masser and Vaaler ("Counting algebraic numbers with large height," Trans. Amer. Math. Soc. 2007), it is easily seen that $\log{N(d)} > Ad$ for $d \gg_A 0$ and any given $A < \infty$.

The mentioned paper of Masser and Vaaler determines the dominant term for the opposite count involving points of large height and bounded degree. The problem considered here, involving bounded height and large degree, is likely to be more delicate. The basic lower bound $\log{N(d)} \gg d$ of the previous paragraph is likely to be improved by taking an optimal $m = m(d)$ and examining the error term (rate of convergence) in the Masser-Vaaler estimate. Perhaps such an improvement could point to the right asymptotics for $\log{N(d)}$?

Has this asymptotic been studied?

The answer. Indeed $\log{N(d)} \asymp d^2$, and this is not hard to see modulo a certain irreducibility hypothesis, which at the very least is implied by a conjecture of Poonen (cf. Thm. 1 in Bertini theorems over finite fields). By the Jensen-Mahler formula, the height $h(\alpha)$ of an algebraic number of degree $d$ with minimal polynomial $f \in \mathbb{Z}[x]$ coincides with the normalized logarithmic Mahler measure: $$ h(\alpha) = d^{-1} \log{M(f)} := d^{-1} \int_{S^1} \log{|f(z)|} \, \frac{d\theta}{2\pi}. $$ By the triangle inequality and a term-by-term integration, this is bounded above by $d^{-1} \log{\ell_1(f)}$, where $\ell_1(f)$ is the $L^1$-norm of the vector of coefficients of $f$. Now of course the inequality $\ell_1(f) < e^d$ has $\exp((1-o(1))d^2)$ solutions $f \in \mathbb{Z}[x]$ of $\deg{f} = d$, and it should not be too difficult to show that as many of them are irreducible (perhaps I am underestimating this problem). This supplies the lower bound $\log{N(d)} \geq (1-o(1))d^2$.

Northcott's bound gives $\log{N(d)} < (1+\log{2})d^2 + O(d)$ in the other direction, and we certainly have $\log{N(d)} \asymp d^2$. Apart from verifying the claim on irreducibility, what remains to be seen is whether equality holds in $\log{N(d)} \geq (1-o(1))d^2$. But I suspect this could be a delicate issue.

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  • $\begingroup$ Which notion of height are you using? $\endgroup$ – Will Sawin Jul 27 '14 at 21:20
  • $\begingroup$ Weil's absolute logarithmic canonical height. E.g., $h(2^{1/d}) = (\log{2})/d$. $\endgroup$ – Vesselin Dimitrov Jul 27 '14 at 21:22
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    $\begingroup$ One can force irreducibility here using congruence conditions. If $f(d)$ is congruent mod $2$ to an irreducible polynomial than it is always congruent. And close to $1/d$ polynomials mod $2$ are irreducible. Because the $L^1$ norm bound is large, different polynomials mod $d$ occur with very similar frequencies, so you should only lose an additive factor of $\log d$ in the exponent - i.e. multiplicatively almost nothing. $\endgroup$ – Will Sawin Dec 11 '16 at 10:04
  • $\begingroup$ @WillSawin: Thanks for this! So this solves the $\log{N(d)} \asymp d^2$ question. The $\sim d^2$ refinement appears to be a lot subtler, though it still seems to me it ought to hold, with the proportionality coefficient of $1$ (?). $\endgroup$ – Vesselin Dimitrov Dec 12 '16 at 2:41
  • $\begingroup$ I think it's not subtle. There are $\exp((1-o(1))d^2)$ solutions, as you say, and I'm saying at least$1/d$ correspond to irreducible polynomials, for a total of $\exp((1-o(1)d^2-\log d)= \exp((1-o(1))d^2))$. $\endgroup$ – Will Sawin Dec 12 '16 at 7:01
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We can make a conjecture based on the function field model. Replace $\mathbb Q$ with $\mathbb F_q(t)$, that is, the function field of $C= \mathbb P^1$. Then any element of $\overline{\mathbb F_q(t)}$ corresponds to a (multivalued) algebraic function on $C$. This is the same as an irreducible subvariety of $\mathbb P^1 \times C$.

We can directly translate Weil's definition of height to this context. As is usual in height theory the definition becomes similar in the function field context: I believe it's the degree of the projection to the other $\mathbb P^1$, divided by the degree of the projection to $C$.

So the count in the function field case is the number of degree $(d-1,d)$ irreducible curves in $\mathbb P^1 \times \mathbb P^1$. Ignoring irreducibility, the number of such curves is $(q^{ d(d+1)} -1 )/(q-1)$. A positive proportion of these are irreducible, because a positive proportion are smooth by Bjorn Poonen's Bertini theorem over finite fields.

So in this case the logarithm of the count is $\Theta(d^2)$. In fact, it is $(\log q+o(1) )d^2$.

So we, perhaps naively, expect the same asymptotic - proportional to $d^2$ - in the number field case.

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  • $\begingroup$ I think there is a factor of $\log q$ I missed in the intersection theory formula for height, which cancels with the $\log q$ in the count. $\endgroup$ – Will Sawin Jul 28 '14 at 1:53
  • $\begingroup$ There is certainly no way that we could get a lower bound of $cd^2$ by using an $m(d)$ in the Masser-Vaaler count that I mentioned. $\endgroup$ – Vesselin Dimitrov Jul 28 '14 at 7:41
  • $\begingroup$ Very nice! So Northcott's bound, apart from the coefficient of $d^2$, might give the right order of magnitude in the number field setting too; I didn't think about this. Do you know what the proportionality constant is, in this function field model? (You use Bertini, and moreover you only need the irreducible curves. Are those of density 1?) $\endgroup$ – Vesselin Dimitrov Jul 28 '14 at 7:42
  • $\begingroup$ I think, though, that there could be problems with this particular analogy. One indication is that there are fewer torsion (height zero) points in number fields than in function fields: $\Theta(d^2)$ and $\Theta(q^d)$, respectively. Consider then the analogous question for points of height bounded by $c/d$, where $c > 0$ satisfies Lehmer's conjecture: same asymptotics. While the counts do match when $h \gg_d 0$, I would not find it too shocking if there were fewer small points in the number field case - despite the notorious difficulty of Lehmer's problem. $\endgroup$ – Vesselin Dimitrov Jul 28 '14 at 10:59
  • $\begingroup$ @VesselinDimitrov I think the count in the function field is of the form $e^{d^2+O(D)}$. So the constant term for the log of the count is $1$. I'm guessing irreducible curves approach density $1$, because the number and complexity of singularities needed to be reducible increases with $d$. However a larger influence is the $+1$ and the rounding if you put a $\log q$ normalizing factor on the height, so that determines the error term. $\endgroup$ – Will Sawin Jul 28 '14 at 15:02
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Dubickas, Algebraic numbers with bounded degree and Weil height, Bull Aust Math Soc 98 (2018) 212-220, writes,

For a positive integer $d$ and a nonnegative number $\xi$, let $N(d,\xi)$ be the number of $\alpha\in\overline{\bf Q}$ of degree at most $d$ and Weil height at most $\xi$. We prove upper and lower bounds on $N(d,\xi)$. For each fixed $\xi>0$, these imply the asymptotic formula $\log N(d,\xi)\sim\xi d^2$ as $d\to\infty$, which was conjectured in a question at Mathoverflow [Asymptotics for algebraic numbers of height less than one ].

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As I indicated above, $\log{N(d)} \asymp d^2$ reduces to a problem about irreducible polynomials, the very likely affirmative answer to which would prove the lower bound $n(h,d) \geq (h-o(1))d^2$, as $d \to_h \infty$, for the logarithm of the number of points in $\mathbb{G}_m(\bar{\mathbb{Q}})$ of degree $d$ and height $< h$. Northcott's upper bound is $n(h,d) \leq (h+\log{2})d^2 + O(d)$.

I nonetheless leave below the remarks I had made.


I do find Will's model to be a convincing evidence that the logarithmic asymptotic count should likewise be $\log{N(d)} = (1+o(1))d^2$. Below I suggest an explanation for the apparent discrepancy in the two models as far as the distribution of points of very small height is concerned. Taking into account this and the (Schanuel-Schmidt)-Masser-Vaaler theorem, I would think that the statistical parallels between number and function fields are much too precise to think of any other answer.

Let $n(h,d)$ be the logarithm of the number of points of degree $d$ and logarithmic height $< h$. Northcott's estimate, $n(h,d) \leq (h+\log{2})(d^2+d) + O(d)$, is valid also in the function field case (where the term $\log{2}$, which arises from the triangle inequality at the archimedean places, is not needed); but there, by an extension of Will's argument, we have a precise asymptotic: Assuming as we may that $h \in \frac{\log{q}}{d} \mathbb{Z}^{> 0}$, then in fact $n(h,d) = h(d^2+d) - 1 + o(1)$ as $\max(h,d) \to \infty$. In other words: in our function field model, Northcott's bound is always sharp.

For the number field case, Masser and Vaaler, extending previous work of Schanuel and Schmidt, consider the count for $d$ fixed and $h \to \infty$; they prove in particular that $n(h,d) = h(d^2+d) + k(d) + o(1)$, with an explicit term $k(d) \asymp d\log{d}$. (Their result is more precise than this, and it applies more generally to relative extensions over a fixed number field). Hence, in this regime, Northcott's logarithmic bound is again sharp.

In $\overline{\mathbb{Q}}^{\times}$, however, in contrast to the situation for our function field model, Northcott's prediction breaks down completely when we take $h = c/d$, where $0 < c < \infty$ is a constant: $\exp(hd^2)$ is then exponential in $d$, whereas we expect $\# \{ \alpha \mid h(\alpha) < c/d, \, [\mathbb{Q}(\alpha):\mathbb{Q}] \leq d \}$ to be polynomially bounded, perhaps even by $\kappa(c)d^{1+\exp(c)}$. (This looks like an interesting problem; does such a bound appear anywhere in the literature?)

Here, I believe, is the explanation for this apparent discrepancy (exponential versus polynomial) in the distribution of points of very small height - the torsion points in particular. There are two things to keep in mind. The first is the distinction between general Weil heights and canonical (dynamical, normalized, Neron-Tate) heights: they differ by a bounded amount and therefore they produce comparable asymptotics - certainly the same rate of growth - for large heights; whereas the distribution of small points is a very fine intrinsic property of the latter heights. The second point to keep in mind is that our particular function field model here is isotrivial (constant, in fact); this accounts for the profusion of torsion (as well small non-torsion) points. An analogy here is to think of a constant elliptic curve over a complex function field: its points of zero canonical height are all the constant sections (an uncountable set), whereas for non-isotrivial elliptic curves they only comprise of the countable set of torsion points.

In either setting, think of our height $h(\cdot)$ as the canonical dynamical height - the global Tate limit attached by Call and Silverman to a dynamical system on $\mathbb{P}^1$ - of the iteration $z \mapsto z^2$. The difference is that in the function field case, this iteration is isotrivial. Isotriviality has no counterpart in the number field setting, and a more accurate function field model in our problem would be to consider the dynamical height $\hat{h}_f$ attached to a non-isotrivial map of $f : \mathbb{P}^1 \to \mathbb{P}^1$ over $\mathbb{F}_q(t)$; or, if you prefer, the canonical height on a non-isotrivial elliptic curve over $\mathbb{F}_q(t)$. In those cases, as in the arithmetic setting, the number of points of degree $\leq d$ and height $< c/d$ should be similarly bounded polynomially in $d$, and the discrepancy in the count will not occur.

In any case, $\hat{h}_f - h = O_f(1)$, and this implies that for $h \gg_ f 0$ sufficiently large, the logarithmic count $n_f$ for $\hat{h}_f$ still satisfies $n_f(h,d) \asymp h(d^2+d)$. (A moment of reflection will show that the proportionality constant should still be $1$, although this could be difficult to prove; all we know at this point is that $n_f(h,d)$ is locked between two positive multiples of $h(d^2+d)$.) But for any fixed positive $h > 0$, we may construct enough points of height $< h$ by taking inverse images under $f^{\circ N}$ of points of height $< h \cdot (\deg{f})^N$, which will be large enough as soon as $N \gg 0$. Hence, for any fixed $h > 0$, we still know that $n_f(h,d) \asymp h(d^2+d)$.

This now should hold in the number field setting too.


Let me finish by adding one more remark about the small point analogy. It is often said that the crux of Lehmer's problem, stating $h(\alpha) > c/d$ unless $\alpha$ is torsion, is in the presence of archimedean places, and that the question is pointlessly trivial in the function field setting. This of course is true for the constant function field model considered in Will's answer, and it is true more generally for polarized dynamical systems with everywhere potential good reduction over a global function field (e.g., abelian varieties with everywhere potential good reduction). However, the real analogy arises when we have degenerations; for instance, for the case of the canonical height on a non-isotrivial elliptic curve over $\mathbb{F}_q(t)$, where I believe that Lehmer's problem would be every bit as difficult as the original one for $\mathbb{G}_m$ over $\bar{\mathbb{Q}}$.

If this is correct then, from a point of view of algebraic dynamics, the essential difficulty in Lehmer's problem is not so much in the failure of the ultrametric triangle inequality in number fields, as it is in the presence of places of degenerating - or chaotic - dynamics (though of course the two points are related). After all, it is a common wisdom in arithmetic geometry that archimedean fibres (or complex spaces) should be regarded as totally degenerate: think of the Tate curve, Mumford's theory of $p$-adic Shottky groups, or the non-triviality of Julia sets in Berkovich analytic spaces (they are always trivial for iterations with good reduction over a non-archimedean local field).

In view of this, all the subtleties in the distribution of small points in $\mathbb{G}_m(\bar{\mathbb{Q}})$ appear to be present also in a non-isotrivial dynamical system on $\mathbb{P}^1$ over $\mathbb{F}_q(t)$, or in a non-isotrivial elliptic curve over a global function field. But we saw that in such a dynamical system, $n_f(h,d) \asymp h(d^2+d)$ for any fixed $h > 0$ (or for any fixed $d$). So the same should persist in the number field setting too - presumably, with proportionality constant equal to $1$.

To sum up, as soon as the height is bounded away from zero, Northcott's bound is pretty sharp. This is something I did not expect at the time of asking this question.

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